It's upto what you are going to achieve. If the first column is NOT NULL, COUNT(*) and COUNT(1) have the same result. If it's NULLable, the result may be different.

USE northwind

GO

SET NOCOUNT ON

SET ANSI_WARNINGS ON

select COUNT(*) FROM dbo.Customers

select COUNT(CustomerID) FROM dbo.Customers

select COUNT(Region) FROM dbo.Customers

select COUNT(FAX) FROM dbo.Customers

Result:

-----------

91

-----------

91

-----------

31

Warning: Null value is eliminated by an aggregate or other SET operation.

-----------

69

Warning: Null value is eliminated by an aggregate or other SET operation.

Their execution plans are slightly different. The last one uses clustered index scan and others use index scan on my machine.

select COUNT(*) FROM dbo.Customers

StmtText                                                                   

---------------------------------------------------------------------------

  |--Compute Scalar(DEFINE[Expr1002]=Convert([Expr1003])))

       |--Stream Aggregate(DEFINE[Expr1003]=Count(*)))

            |--Index Scan(OBJECT[Northwind].[dbo].[Customers].[Region]))

select COUNT(CustomerID) FROM dbo.Customers

StmtText                                                                   

---------------------------------------------------------------------------

  |--Compute Scalar(DEFINE[Expr1002]=Convert([Expr1003])))

       |--Stream Aggregate(DEFINE[Expr1003]=Count(*)))

            |--Index Scan(OBJECT[Northwind].[dbo].[Customers].[Region]))

select COUNT(Region) FROM dbo.Customers

StmtText                                                                        

--------------------------------------------------------------------------------

  |--Compute Scalar(DEFINE[Expr1002]=Convert([Expr1003])))

       |--Stream Aggregate(DEFINE[Expr1003]=COUNT_BIG([Customers].[Region])))

            |--Index Scan(OBJECT[Northwind].[dbo].[Customers].[Region]))

select COUNT(FAX) FROM dbo.Customers

StmtText                                                                                   

-------------------------------------------------------------------------------------------

  |--Compute Scalar(DEFINE[Expr1002]=Convert([Expr1003])))

       |--Stream Aggregate(DEFINE[Expr1003]=COUNT_BIG([Customers].[Fax])))

            |--Clustered Index Scan(OBJECT[Northwind].[dbo].[Customers].[PK_Customers]))